Lecture 12
Root locus
Motivation
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(performance via “dominant poles” design)
Given a combined requirement on overshoot, rise time and settling time, we could get:
If we place dominant poles of the closed
loop in the white region of the complex plane, the system would have the desired transients in
step response
Brown colour represents a “forbidden region”.
Motivation:
(simplest proportional controller)
Consider the closed-loop system, where K is a parameter (i.e. “proportional” controller)
Design question: Can we select K so that the poles are in the desired region?
Motivation:
Transfer function of the closed loop system:
Locations of closed loop poles depend on the parameter K:
Motivation (proportional controller)
Plot the poles for this system for K>0
for all possible values of K>0. We have:
Motivation
We obtain a plot in complex plane
Open loop poles
of root locus
This is a “root locus” for the given system.
A typical design question:
Can all poles be “placed” into the desired region of complex plane by choosing K?
Root locus
Complex domain (pole locations)
Motivation:
arbitrary parameter in the controller
Characteristic equation is:
We will consider this example in detail later.
Motivation: robustness analysis
Characteristic equation is:
Problem formulation:
Plot in the complex s plane the locations of all roots of the equation
as K varies from 0 to infinity.
This plot is called the (positive) “root locus”.
Phase and magnitude conditions
Note that if a point in the complex plane lays on the root locus, it has to satisfy
which implies that these conditions hold:
Phase condition
Since K is positive, the phase depends only on poles and zeros of F(s). In other words, for any point on the root locus, we have:
Graphical interpretation
Calculate K for a point on root locus
The root locus is parameterized with the gain K>0.
If we want to calculate the value of K that corresponds to a specific point on the root locus, we can use the gain condition:
Sketching root locus via Matlab
User can define a transfer function and then use the command “rlocus” to plot its root locus.
To understand Matlab plot, it is useful to learn how to sketch root locus by hand.
It is useful to use the command “sgrid” to get a grid of lines with constant damping and constant natural frequencies.
Root locus via Matlab
>> rlocus([1],[1 3 2 0]) >> sgrid
Lines of constant damping
and constant natural frequency are ploted using “sgrid”
Main features of root locus
Number of branches
Open loop poles (starting points for K=0)
Open loop zeros (limiting points for K infinity) Parts of real line that belong to root locus
Asymptotes
Breakaway point (branches intersect)
Intersections with imaginary axis
Angles of departure or arrival at poles/zeroes
Open loop poles/zeroes
Open loop poles are points where branches of root locus start from (small K).
Open loop zeroes are points where some branches of root locus converge to (large K).
3 open loop poles (crosses) No open loop zeroes (circles) 3 branches
(blue, green, red)
Parts of real axis that belong to locus
Thick red line denotes
parts of real axis that belong to root locus
These parts of real axis belong to root locus.
Asymptotes of root locus
Red lines denote 3 asymptotes Asymptotes are determined by:
– Point where they intersect the real axis – Angle with the positive real axis
Breakaway point
Breakaway point where two branches cross on the real axis – repeated poles.
Sometimes we have repeated complex poles. This is where several branches intersect.
Intersections with imaginary axis
Intersections with imaginary axis.
Angles of departure/arrival
rlocus([1 3 -1 0 1],[1 2 -2 10 4 4])
This system can not be stabilized by choosing K!
Angle of departure
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